Efficient and accurate spectral analysis of large scattering problems using wavelet and wavelet-like bases

Authors


Abstract

[1] This paper presents a novel technique for the very efficient and accurate two-dimensional (2-D) spectral analysis of large cylindrical scatterers with arbitrary geometries using the wavelet and wavelet-like transform. When such complex problems are solved through an integral equation (IE) technique combined with the method of moments (MOM), a linear equation system with a huge number of unknowns needs to be solved for each incident field. In this work, wavelet bases are successfully proved to expand the integral operator in a very sparse matrix form, thus providing large savings on computational costs and memory storage requirements. Furthermore, the use of the spectral formulation allows us to characterize the scattering behavior of an object for any possible incidence, thus drastically reducing the number of linear equation systems to be solved. Comparative benchmarks between the new wavelet technique and the original approach for solving the scattering of circular cylinders and strips are presented for different orthogonal families: locally supported Daubechies bases, orthogonal quasi-symmetric Coiflets bases, and wavelet-like bases. Then the efficient wavelet technique is applied to the accurate characterization of more complex geometries, such as large square cylinders.

1. Introduction

[2] IEs have been traditionally employed for the accurate solution of a wide variety of electromagnetic problems, such as the analysis of scatterers solved in the work of Poggio and Miller [1973], the analysis and design of boxed multilayered circuits described by Eleftheriades et al. [1996], and the efficient modal computation of arbitrarily shaped waveguides shown in the work of Conciauro et al. [1984]. These integral equation problems are usually solved following the well-known MOM described by Harrington [1968]. When such approach is applied to either electrically large structures, highly accurate applications, or objects with many small geometry details, matrix problems with a very large number of unknowns arise.

[3] To avoid the aforementioned drawback related to the IE-MOM approach, and take profit of the accuracy and efficiency advantages of such technique, the use of the wavelet transform has been recently proposed [see Sarkar et al., 2002]. It was initially applied to microstrip circuits by Wang and Pan [1995] and Sabetfakhri and Katehi [1994], to the efficient waveguide mode computation by Wagner et al. [1993], and to the analysis of double-slot apertures in planar conducting screens by Steinberg and Leviatan [1993]. In the recent years, the application of the wavelet transform to the efficient analysis of 2-D electromagnetic scattering problems has become rather popular, as can be seen in the work of Steinberg and Leviatan [1994], Wagner and Chew [1995], Baharav and Leviatan [1996a, 1996b], Nevels et al. [1997], Deng and Ling [1999], Baharav and Leviatan [1998], Golik [2000], and Guan et al. [2000]. For instance, Steinberg and Leviatan [1994] proposed a periodic wavelet expansion to overcome the classical problem of expanding a given function of finite support. An extensive comparison of wavelets with other fast algorithms for solving the cited 2-D problem can be found in the work of Wagner and Chew [1995], where it is concluded that the use of wavelets in small problems is not very advantageous. Baharav and Leviatan [1996b], Deng and Ling [1999], and Baharav and Leviatan [1996a, 1998] proposed novel approaches for the incorporation of wavelet expansions into integral equations, which are mainly based on the impedance matrix compression using either adaptively constructed, or iteratively selected, wavelet basis functions. These wavelet transforms are computationally very expensive and they have not been selected for further study in this paper. Different wavelet bases for solving IEs were first studied in the work of Nevels et al. [1997], where semiorthogonal compact support bases outperformed infinite support orthogonal ones. However, Golik [2000] and Guan et al. [2000] studied the most suitable wavelet bases for scattering problems, and they conclude that, in such case, the orthogonality of wavelets is a crucial issue to be fulfilled. Another research direction is to use smooth local cosine bases instead of wavelet bases. This is proposed in the work of Sweldens [1996] and applied to scattering analysis in the work of Pan et al. [2003]. However, if no a priori information about the scatterer is known, many processing parameters should be adjusted in search for the best matrix compression and that increases consequently the computing cost.

[4] It is also convenient to remark that most of the works on scattering problems cited in the previous paragraph have been mainly focused on small 2-D scatterers, whose cylindrical conducting contours have been coarsely segmented. Such approach leads to small matrix problems, where the advantages of the wavelet approach are clearly lessened. Furthermore, the MOM solution followed in such references has been always formulated in the spatial domain, thus leading to incidence-dependent solutions, which are not very suitable for the later consideration of multiple scattering problems. In those methods, the solution of the scattering problem for different incidences involves the solution of a linear equation system for each incident field. In this paper, a formulation that computes the solution of the scattered field for any incidence solving only a limited set of linear equation systems is followed [see Esteban et al., 1997].

[5] Despite of all the previously commented efforts, much research on the wavelet transform, and its practical application to the accurate solution of electromagnetic scattering problems, is still needed. Specially, wavelet-like bases have not been sufficiently studied in the frame of 2-D scattering problems compared to other popular wavelet bases such as Daubechies. Therefore, taking into account the present state-of-the-art reached in this research area, it seems appropriate to face the combination of wavelet and wavelet-like bases and the new spectral domain technique for the efficient and accurate solution of large MOM matrix problems with oscillatory kernels. If the application of wavelet theory in this new scenario gave good enough results, a further step for the very efficient and accurate solution of large scattering problems by personal computers without special memory arrangements would be given. The improvement of the efficiency related to the use of the wavelet transform for solving large 2-D scatterers would encourage its use in more complex 3-D scattering scenarios.

[6] This paper is organized as follows. First, the 2-D electromagnetic scattering problem under consideration is formulated and solved in the spectral domain. Next, the wavelet transform is successfully introduced for the compression of the corresponding impedance matrix. After testing several wavelet and wavelet-like families, the best ones have been selected to perform a comparative study included in this work. Such study has been realized through simple 2-D large scattering problems, such as circular and square cylinders, as well as strips. Very accurate numerical results for the scattering behavior of such objects are presented, and the efficiency gain provided by the use of wavelets is discussed. Finally, the main conclusions derived from the detailed study performed in this paper are briefly outlined.

2. Theory

[7] The electromagnetic problem to be solved is to fully characterize the scattering response of perfectly electric conducting cylinders (invariant in z axis) with arbitrary cross section when any transverse magnetic (TM) wave polarized along z axis (TMz) excites it in open space.

[8] In this section, we first present the basics of the spectral and MOM techniques which will be used to solve our problem. Next, we briefly describe how the wavelet transform can be used to improve the efficiency of such technique. Then the wavelet and wavelet-like bases employed in this study, as well as their practical implementation, are introduced. Finally, the procedure for evaluating the efficiency improvement related to the use of the wavelet transform is described.

2.1. Basics of the Spectral MOM Technique

[9] To solve the scattering problem, the following modal decomposition of incident Ei and scattered electric fields Es using cylindric modes, described in the work of Esteban et al. [1997], is applied:

equation image
equation image

where (ρ, ϕ, z) are the coordinates of an observation point ρ in the selected cylindrical system, k is the free-space wave number, Jp and Hq(2) are, respectively, the pth-order Bessel function and the qth-order Hankel function of the second kind, and ip and cq are the corresponding incident and scattered spectral coefficients. With regard to the number of modal coefficients, Ni and Nd, required to reconstruct accurately the incident and scattered fields, they can be determined following the practical criteria proposed by Esteban et al. [1997, 2002].

[10] The main aim of the spectral technique is to fully characterize the scattering behavior of any electrical conducting object by means of a generalized scattering matrix, D, which will provide the scattered field by the object under any arbitrary incidence as follows

equation image

[11] The generalized scattering matrix elements (dqp) can be analytically known for some canonical geometries (i.e., a circular cylinder), as can be seen in the work of Balanis [1989]. However, for arbitrary geometry objects, a numerical strategy based on MOM will be followed. Next, we offer a brief description of such strategy in order to better understand the new contents of this paper. More technical details can be found in the work of Esteban et al. [1997, 2002].

[12] The unknown longitudinal current induced on the arbitrary scatterer conducting surface due to the pth incident mode equation image must be first computed. To do so, the following electric field integral equation (EFIE) outlined by Harrington [1968] is first established:

equation image

where equation image is the longitudinal component of the pth incident field mode, C is the contour of the scatterer cross section defined by the l′ parameter, and ρ(l′) is a vector pointing to such contour. In order to solve this EFIE, a classical MOM technique in the point-matching variant described by Harrington [1968] and Peterson et al. [1998] has been used, thus obtaining the following algebraic linear equation system:

equation image
equation image
equation image

where ΔCn is the nth interval in which the whole contour C of the scatterer has been segmented. To implement the MOM technique, pulse basis functions centered at ρn (n ∈ [1, …, N]) points have been used, and ρm (m ∈ [1, …, N]) indicates the center of each segment where the Dirac Delta functions are placed. It must be noticed that the diagonal elements of the Z matrix (Zmm terms) do involve a singular integral, but approximated expressions for such diagonal elements, as well as for the nonsingular elements (Zmn with mn), can be found in the work of Harrington [1968].

[13] If the previous MOM approach is repeated for all possible incident spectral modes (p ∈ [−Ni, …, Ni]), a global matrix system is finally obtained

equation image
equation image

[14] In conventional 2-D scattering problems, the solution of one of such linear equation systems is needed for each possible incident field, and therefore the computation cost grows with the number of incident fields to be considered. In our approach, equation (8) represents a set of 2Ni + 1 linear equation systems, which do share the same impedance matrix Z and fully characterize any possible TM incident field. Consequently, it should be pointed out that the new spectral formulation drastically decreases the computation time compared to the classical formulation. Anyway, the effort related to the solution of equation (8) clearly dominates the global computational effort. Once matrix I is calculated, the generalized scattering matrix can be easily found as follows

equation image

where S is a transformation matrix based on the Bessel addition theorem [see Harrington, 1961, 1968].

[15] In order to evaluate the accuracy of this method, the square boundary condition error ΔEbc (%) defined by Baharav and Leviatan [1998] has been employed. Such error takes into account the degree of accomplishment of the boundary condition along N0 points of the scatterer contour in the following way

equation image

where Ezs is the scattered field by the object under an arbitrary incident wave defined by Ezi. For all the examples considered in this paper, a very low threshold value (0.05%) has been chosen for ΔEbc, which guarantees a high degree of accuracy for most real applications. The choice of such threshold value directly affects the number of discrete points N related to the MOM implementation. Therefore, following this error criterium, a linear equation system with a large number of unknowns will have to be solved. Such large linear systems will also appear when solving the scattering behavior of either electrically large objects or scatterers with very small geometric details. In the next paragraphs we propose the use of the wavelet transform to alleviate the CPU cost related to the solution of the aforementioned large linear systems.

2.2. MOM Implementation Using the Wavelet Transform

[16] The solution of the linear equation systems in equation (8) can be solved by two main techniques. The first one is a direct solver, based on Gaussian elimination or LU decomposition, that achieves a solution in O(N3) operations. The second approach is to use a recursive method, in which each iteration involves matrix-vector products taking O(N2) operations. That solution is usually avoided because the total number of iterations for solving a matrix equation depends on the matrix condition number. Consequently, iterative techniques are not extensively employed in the case of large dense matrices, unless some suitable preconditioning schemes are found, as described by Heldring et al. [2002]. Furthermore, both techniques do require high computational efforts and management of large memory blocks. Therefore the discrete wavelet transform can improve the overall computation cost and memory requirements by turning the impedance matrix Z into a sparse one. The challenge when applying this technique comes from the fact that wavelet transform for integral operators compression is well suited for large smooth real systems of equations, as can be seen in the work of Alpert et al. [1993] and Beylkin et al. [1991]. However, oscillatory kernels, like the Hankel function that appears in our problem, have not been so well characterized through wavelet bases up to now.

[17] Then the main goal of this work is to reduce the computation cost and the memory requirements in the solution of the matrix equation (8) using the wavelet transform, whereas high accuracy is kept. For that purpose, several wavelet families have been considered, and three bases have been finally selected for a deeper analysis: the well known compactly supported orthonormal Daubechies bases described by Daubechies [1988], Coiflet quasi-symmetric orthonormal bases described by Daubechies [1992] and Sarkar et al. [2002] and the wavelet-like scheme shown by Alpert et al. [1993] and Chui [1991]. To the authors knowledge, the wavelet-like scheme has been only proposed in the work of Wagner et al. [1993] for solving waveguide mode computation problems. Consequently, the use of wavelet-like bases in the frame of scattering problems with oscillatory kernels is first considered in this paper.

[18] In all the selected wavelet bases, the transformed integral operator presents a diagonal banded form usually known as finger structure. After applying a threshold, the operator is converted into a very sparse matrix. Then very efficient special purpose recursive techniques for sparse matrices can be applied, whose computational burden is in the order of the nonzero matrix entries. Nevertheless, the wavelet approach involves a matrix transformation, whose cost should be included within the global computation budget. So, the benchmark proposed in this work is to reduce the total number of operations including forward and inverse wavelet transform for a certain large number of unknowns when compared to the direct approach.

[19] The conversion of the discrete integral operator into the new wavelet bases is equivalent to carry out a similarity transformation defined by a real and unitary matrix W, whose rows contain the new orthogonal wavelet or wavelet-like bases as described by Steinberg and Leviatan [1993]. The original problem is now represented in the wavelet domain as follows

equation image

where W is the transformation matrix, superscript T stands for the transpose of a matrix, and Zt, It and Vt are, respectively, Z, I and V expressed in the new bases. This transformation method is called tensor product or standard form wavelet transform. Owing to the wavelet bases properties, Zt presents many elements that are close to zero. However, they should be zero to become a sparse matrix. So, the application of a threshold is necessary, which is a key step related to the desired accuracy and efficiency gain. The threshold value τ is given in percentage units, and it is normalized to the maximum absolute value of the matrix entry. In our study, we have implemented the threshold operation through a simple comparison of the absolute value of the complex elements, and sparsity is defined as the percentage of matrix zero entries after thresholding.

2.3. Wavelet and Wavelet-Like Bases Description

[20] Functions ψ and ϕ are, respectively, the wavelet function and its corresponding scaling function described by Daubechies [1992]. They are defined in the frame of the wavelet theory and can be constructed with finite spatial support under the following conditions

equation image
equation image
equation image

where L is the length of the wavelet filters whose coefficients are hk and gk, and k is the corresponding spatial index. In this study, only orthonormal bases with corresponding hk and gk with the same finite support are studied. The scaling function must also satisfy equation (15) for normalization purposes. The wavelet filters are related by the following relationship that turns them into a set of quadrature mirror filters, being hk a low-pass filter and gk its corresponding high pass filter

equation image

[21] The wavelet family presents M vanishing moments if the following expressions related to the wavelet function ψ and its corresponding high pass filter gk apply

equation image
equation image

[22] Then the translated and dilated wavelet functions form the following set of bases

equation image

where j is a positive integer representing a certain scale level. Starting from a decomposition level j, the corresponding dilated jth scaling function, the dilated jth wavelet function and the rest of dilated and translated wavelet functions at finer levels also form a set of orthogonal bases, which are used in their discrete version to form the rows of matrix W.

[23] The wavelet family may be designed with several degrees of freedom to get different properties, such as finite spatial support, symmetry, orthogonality, a certain number of vanishing moments, frequency localization and a level of regularity. However, not all these features may be simultaneously achieved as shown by Daubechies [1992]. In this paper, the most important properties for our problem have been identified as orthogonality, compact spatial support and high number of vanishing moments. First, orthogonality is important to preserve the matrix conditioning properties as explained by Golik [2000] and Guan et al. [2000]. If orthogonality is given, only slight variations in the operator condition number would arise due to the non linear threshold operation. Second, the compact support is interesting to improve the transformation efficiency since the computational cost is proportional to the spatial support. Finally, a high number of vanishing moments is important to increase the compression property of the wavelet transform. However, local support and number of vanishing moments are in inverse dependence. For that reason, the best compression ratio in this study is provided by the wavelet-like scheme, because such bases give the highest moment order with the lowest local support.

[24] The well-known Daubechies orthonormal family of compactly supported bases presented by Daubechies [1992] is selected in first place since these bases provide excellent moment order and some degree of regularity. For that reason, they characterize very accurately rapidly varying functions under some constrains, which is the case for the characterization of smooth and oscillatory integral operators [see Daubechies, 1992; Beylkin et al., 1991]. However, their main drawback is that they present very poor symmetry properties. Another orthogonal family considered in this study is the one called Coiflets, which is also described by Daubechies [1992]. Such wavelet bases are quasi-symmetric at the cost of larger spatial support, and they have been considered in order to test the symmetry effects in our study. For the sake of comparison, Daubechies quadrature filters present M vanishing moments with filter length L equal to 2M, and Coiflets quadrature filters provide M vanishing moments with filter length L equal to 3M.

[25] Both wavelet families provide overlapping base support at the same resolution level. As a result, some bases spill out of the interval [1, N] on which the contour is originally defined. Therefore the spatial support of some bases has to be extended with its N periodic version, which can affect the sparsity of the transformed matrix. Another constraint is given by the number of unknowns in the problem, which should be a power of two in order to be able to use the hierarchical fast transform.

[26] Another interesting approach considered in this study is the wavelet-like transform, which was originally introduced by Alpert et al. [1993]. This technique does not lie in the wavelet transform scheme, but it shares some of its properties. The wavelet-like bases are orthonormal, they present local support and all of them have at least M vanishing moments except for M bases. In this case, the dimension of the transformation matrix N should be equal to M · 2n for some integer n, and no periodic extension is needed thanks to the bases construction procedure.

2.4. Efficient System Basis Transformation

[27] The transformation of the dense matrix into the discrete wavelet domain is a critical step in this method. If equation (12) is directly applied using matrix-matrix products, no gain is achieved since the transformation process itself involves O(N3) operations. Obviously, this computation cost is comparable to the order of the original direct solver, and some efficient transformation method should be implemented in order to improve the overall efficiency. Fortunately, the wavelet transform can be accomplished by a different method, which needs less computation cost. Using this new fast wavelet transform, the system transformation can be done in O(N2) operations [see Wagner and Chew, 1995], which is low compared to the original O(N3) operations.

[28] Such fast wavelet transform does only apply to wavelet families with finite quadrature filters, which is the case of Daubechies and Coiflets. In these cases, the fast wavelet transform is accomplished by filtering and decimating (see Sarkar and Kim [1999] for details). At the coarsest level, the vectors of N elements are filtered by hk and gk employing 2LN multiplications for each vector, where L is the filter length. At each iteration, the number of required operations is divided by two with regard to the previous stage, due to the data decimation. Therefore the number of multiplications per each vector transformation is

equation image

where N rows and N columns must be processed. Consequently, the overall number of multiplications is given by 4LN(2N − 1), and the 2-D transform is made in O(N2) operations.

[29] Alternatively, the wavelet-like implementation is performed following the tensor product approach described by equation (12) since matrix W used for the wavelet-like transform is sparse [see Alpert et al., 1993; Chui, 1991], and matrix-matrix products may be carried out with MN2(1 + log2equation image) multiplications. The 2-D transform is achieved in 2M N2(1 + log2equation image) multiplications, and the whole process also presents a computation cost of order O(N2) operations, as in the previous case, but with a higher constant factor.

[30] For both transformation methods, the number of vanishing moments is a key parameter. It is clear that an increment of the vanishing moment order would characterize the matrix with greater accuracy and higher matrix sparsity, but, at the same time, it would also increase the number of operations in both implementations. Therefore a trade-off between filter length and efficiency must be assumed in order to find the optimum wavelet or wavelet-like order.

[31] To conclude, it must be noticed that the cost of the system bases transformation just described will domain in the whole process when N increases [see Guan et al., 2000] since the computational burden of the sparse iterative solver can be reduced to a function of the number of nonzero elements, which is low due to the wavelet transform and threshold application.

2.5. Computational Cost Evaluation

[32] In our efficiency study, a comparison between the original method and the new wavelet approach has been carried out. The procedure that has been measured in the original problem includes the computation of I, sized N × (2Ni + 1), in the matrix equation (8) using a direct solver based on Gaussian elimination for the number of unknowns N given by the desired accuracy and the scatterer size. In the case of the wavelet approach, the procedures that have been taken into account in the computation budget are the wavelet transform of matrix Z in both dimensions, the wavelet transform of matrix V in one dimension, the solution of the system in equation (12) using a sparse solver and the wavelet inverse transform of It to enter the original algorithm chain at the same step. It must be noticed that the number of unknowns grows in the case of the wavelet approach since N must be a power of two equal or greater than the previous original bound. This fact has been used to enhance the accuracy degraded by the threshold operation. Obviously, this effect has been taken into account in the computational budget just described, which has been measured using the standard command provided by the commercial software Matlab©.

[33] The construction of matrix Z has not been taken into account for comparison, because it is a common factor in all procedures. Besides, it has been proved that it does not affect the results even in the case when N is increased to a power of two.

3. Results and Discussion

[34] The different scattering geometries considered in this work are circular and square cylinders, as well as strips. In such problems, the square boundary condition error ΔEbc has been always kept below 0.05% as explained in section 2.1. The choice of such error condition has required to use at least 72 points per contour wavelength in the case of the circular cylinder, and 100 points per wavelength in the case of the strip and the square cylinder. The previous figure is needed in order to meet the desired error in the original problem. However, when the wavelet transform is applied, the number of points grows to a power of two as explained before, and the threshold value threshold value (τ) is chosen equal to 0.1% since it has been proved that such combined values also keep the error ΔEbc under the targeted limit.

[35] Two kinds of comparative studies among the selected wavelet families have been performed. Since the wavelet transform is the dominant factor in the efficiency budget, a first approach consists on constructing the wavelet bases with approximately the same spatial support, i.e., the same number of nonzero entries in the W matrix. To accomplish such condition, the Daubechies bases with filter length L equal to 6 (M = 3), the orthogonal Coiflets with filter length equal to 6 (M = 2), and wavelet-like bases with M equal to 4 are selected. These particular values of filter lengths and consequent moment orders are chosen because they offer the best results for the original problem segmentation. A second kind of comparative study is performed among wavelets with the same number of vanishing moments, which has allowed to identify the wavelet family with a lower transformation cost.

[36] The first example considered is the circular cylinder. The structure of the transformed impedance matrix Zt for a circumference equal to 6.3λ and N equal to 512, using wavelet-like and wavelet transform, are shown in Figure 1. As it can be observed, the expected finger structure of wavelet transformed integral operators has been obtained in both cases.

Figure 1.

Circular cylinder: normalized absolute value of Zt for a cylinder of circumference equal to 6.3λ (N = 512) after applying fourth-order wavelet-like bases and 6-length Daubechies bases with threshold τ equal to 1%.

[37] For the circular cylinder case, the results for the first kind of comparative study mentioned above can be seen in Figures 2, 3, and 4 for different circumference contours (from 3.14λ to 28.27λ). As seen in Figure 2, the error ΔEbc is mostly kept below the upper limit as required, even though there are some particular cases in which Coiflets and Daubechies do not satisfy such requirement. The error oscillatory nature observed in Figure 2 and the rest of figures is produced because the number of unknowns grows proportionally to the scatterer size, but in the wavelet case it is increased to the next power of two. As mentioned previously, extra segmentation compensates for the threshold error. However, when the original matrix size reaches up to a power of two, the error gets higher consequently because in that case the number of unknowns is the same for the original and the transformed problem. In Figure 2, it can be also noticed that the wavelet-like scheme always provide the lower error values, which can be attributed to its higher number of vanishing moments. Consequently, the sparsity is clearly higher (around 99%) in the case of the fourth-order wavelet-like bases (see Figure 3).

Figure 2.

Circular cylinder: Square boundary condition error for different bases with approximately the same spatial support using a threshold equal to 0.1%.

Figure 3.

Circular cylinder: sparsity degree for different bases with approximately the same spatial support using a threshold equal to 0.1%.

Figure 4.

Circular cylinder: number of floating point operations for different bases with approximately the same spatial support using a threshold equal to 0.1%.

[38] In the previous study, the number of millions of floating point operations (Mflops) gets rather lower than the original problem for all cases, as it is graphically shown in Figure 4, where it can be observed a quadratic behavior with N in comparison with the cubic dependance of the original problem. That is due to the fact that the sparsity gets very high for the three wavelet families, and the sparse solver achieves the solution with very low cost. As it can be seen in Figure 4, the efficiency improvement increases as the scatterer becomes larger. Although all the wavelet families get the main objective of this study, wavelet-like bases characterize more accurately the impedance matrix because they present a higher number of vanishing moments (M equal to 4) for the same spatial support (see Figures 2 and 3). This is also confirmed by the sustained form of the Mflops curve in Figure 4 for such wavelet-like bases in the interval of radius between 2.5λ and 4.5λ. Such behavior means that the dominating operation is the wavelet-like transform, and the sparse solver hardly contributes to the number of operations in the wavelet-like case. However, the drawback of wavelet-like bases comes from the fact that its transform is performed in a different way, which usually demands a slight larger number of operations than the Daubechies and Coiflets transforms, as explained before in Section 2.4.

[39] For the case of the circular cylinder, we have also performed the second kind of comparative study described before, which essentially consists on using the same amount of vanishing moments (M = 4) for the three families. In this study, similar errors and sparsity values have been obtained for all wavelet bases, and for that reason, no graphic results for such parameters are presented. However, a better efficiency should be obtained for the wavelet-like case, which is not fully confirmed by the results shown in Figure 5. This can be attributed again to the higher cost related to the different transformation procedure implemented with the wavelet-like bases.

Figure 5.

Circular cylinder: number of floating point operations for different bases with the same number of vanishing moments using a threshold equal to 0.1%.

[40] From the previous analysis, it can be concluded that both Daubechies and wavelet-like bases are good candidates to be used with the MOM spectral approach for solving large scattering problems, both from the accuracy and efficiency point of view. Another feature that can be extracted from this work is that the quasi-symmetry property introduced by the Coiflets family scarcely improves the results provided by Daubechies bases.

[41] In order to confirm the accuracy provided by the wavelet approach, the induced current on a circular scatterer of circumference 6.3λ under a plane wave incidence with ϕ = π/4 has been computed. Figure 6 successfully compares the results obtained using the wavelet-like bases with those provided by the original MOM spectral approach. The small spikes in the current distribution come from the threshold application. That feature is common to all the studies that apply wavelet transform and thesholding to similar problems [see Steinberg and Leviatan, 1994; Baharav and Leviatan, 1996b; Nevels et al., 1997; Deng and Ling, 1999; Golik, 2000]. However, errors in current distribution do not affect very severely either the accomplishment of the boundary conditions or the far field radar cross section.

Figure 6.

Comparison of the induced current equation image when a plane wave with ϕ = π/4 excites a circular cylinder of circumference equal to 6.3λ for the original method (solid line, N = 450) and for the fourth-order wavelet-like accelerated procedure (dotted line, N = 512). The contour parameter origin is located at (ϕ = 0).

[42] A comparative study using bases with similar spatial support has been performed for a strip with different lengths (from λ to 20λ). From Figures 7 and 8, the same conclusions for the error and efficiency improvement reached in the previous geometry can also be derived for the strip scatterer.

Figure 7.

Strip: Square boundary condition error for different bases with approximately the same spatial support using a threshold equal to 0.1%.

Figure 8.

Strip: number of floating point operations for different bases with approximately the same spatial support using a threshold equal to 0.1%.

[43] Finally, the square scatterer shown in Figure 9 has been accurately and efficiently solved with the wavelet-like proposed family. The error values and the sparsity degree reached for several contour lengths are collected in Table 1. Such results are in good agreement with the previous comments made for the circular and the strip cases. Additionally, the comparison of the current induced in a square cylinder of perimeter 2λ by an incident plane wave with ϕ = π/4, obtained by the original method and the wavelet-like accelerated method, is shown in Figure 9. These results allow to conclude that the wavelet technique proposed in this paper can be successfully applied to a wide range of geometries, without significative alteration of its performance.

Figure 9.

Comparison of the induced current equation image when a plane wave with ϕ = π/4 excites a squared cylinder of perimeter equal to 2λ for the original method (solid line, N = 200) and for the fourth-order wavelet-like accelerated procedure (dotted line, N = 256). The contour parameter origin is located at (x = −a/2, y = −a/2).

Table 1. Square Scatterer: ΔEbc and Sparsity Degree Using Fourth-Order Wavelet-Like Bases and τ = 0.1% for Different Scatterer Contour Lengths (4a) Shown in Figure 9
 OriginalFourth-Order Wavelet-Like
Contour LengthΔEbc, %ΔEbc, %Sparsity, %
1 λ0.03860.023193.8599
2 λ0.02980.018696.7194
4 λ0.02540.016298.2239
8 λ0.02450.016099.0074
12 λ0.02410.009499.5254
16 λ0.02390.015499.4230
20 λ0.02320.023499.2635

4. Conclusions

[44] This paper presents the application of recently proposed wavelet and wavelet-like bases to the spectral method of moments (MOM) characterization of large scattering problems with high accuracy. In such problems, the size of the involved matrices is usually very large. This drawback is alleviated by changing the traditional spatial MOM bases into a new set of wavelet or wavelet-like functions. Furthermore, the use of a spectral formulation allows to characterize the scattering problems under any possible incidence by solving a reduced number of linear equation systems. In order to select the most suitable wavelet bases for such scattering problems, a careful trade-off in search of the optimum parameters like moment order and threshold choice has been investigated. The spectral scattering characterization of several geometries, such as circular and square cylinders as well as strips, has been solved using three different wavelet and wavelet-like families. Computational gain has been obtained for all the selected wavelets with no accuracy loss. However, the best compression ratio has been provided by the wavelet-like and the Daubechies bases. From these successful results, we conclude that wavelet bases combined with the spectral MOM approach offer a simple and extremely efficient method for the accurate solution of large electromagnetic scattering problems. Their main advantages are important savings in computation time and memory storage. However, ever greater gains could be achieved if the wavelet transform, which now dominates the computational effort, could be improved in terms of efficiency.

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